Quantum advantage for differential equation analysis

Bobak T. Kiani,1∗ Giacomo De Palma,2,5 Dirk Englund,1,2,3

William Kaminsky, Milad Marvian,4 Seth Lloyd2,5

1Department of Electrical Engineering and Computer Science, Massachusetts Instituteof Technology, Cambridge, MA 02139

2Research Laboratory of Electronics, Massachusetts Institute of Technology,Cambridge, MA 02139

3QuEra Computing, Boston, MA 021434Department of Electrical and Computer Engineering, Center for Quantum

Information and Control (CQuIC),University of New Mexico, Albuquerque NM 871315Department of Mechanical Engineering, Massachusetts Institute of Technology,

Cambridge, MA 02139

∗To whom correspondence should be addressed; E-mail: bkiani@mit.edu.

Quantum algorithms for both differential equation solving and for machine

learning potentially offer an exponential speedup over all known classical algo-

rithms. However, there also exist obstacles to obtaining this potential speedup

in useful problem instances. The essential obstacle for quantum differential

equation solving is that outputting useful information may require difficult

post-processing, and the essential obstacle for quantum machine learning is

that inputting the training set is a difficult task just by itself. In this paper,

we demonstrate, when combined, these difficulties solve one another. We show

how the output of quantum differential equation solving can serve as the input

for quantum machine learning, allowing dynamical analysis in terms of princi-

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pal components, power spectra, and wavelet decompositions. To illustrate this,

we consider continuous time Markov processes on epidemiological and social

networks. These quantum algorithms provide an exponential advantage over

existing classical Monte Carlo methods.

Introduction

One of the primary proposed applications of quantum computers is the solution of linear dif-

ferential equations on high-dimensional spaces. The ability of quantum computers to represent

N -dimensional vectors as the state of log2N qubits, and to perform linear algebraic transfor-

mations of those states in time poly(logN), then translates into a potential exponential speedup

over classical algorithms for solving such high-dimensional differential equations. The output

of the quantum computer presents the solution to the equation as a quantum ‘history state’ that

is a superposition of the solution at different points in time. The problem then arises: How do

we extract useful information from that quantum solution state? We can measure the expecta-

tion value of different quantities of interest: however, in the example of Markov chains, such

expectation values can often be evaluated efficiently classically via Monte Carlo techniques. To

obtain a quantum advantage that reveals essential features of the solution to the linear differen-

tial equations, we need to perform quantum post-processing on the history state.

Quantum machine learning algorithms for the analysis of data provide potential exponen-

tial speedups over classical machine learning algorithms for methods such as high-dimensional

regression, principal component analysis, and support vector machines [1–4]. A basic problem

with such quantum machine learning algorithms is that the input to the algorithm is a quantum

state that encodes the classical data, and to construct such a state requires the implementation

of a large-scale quantum random access memory (qRAM), a difficult technological task. The

central observation of this paper is that the problem with quantum linear equation solvers –

2

they give quantum states as output – and the problem with quantum machine learning algo-

rithms – they require quantum states as input – effectively solve each other: the output from

the quantum linear equation solver can be used as the input to the quantum machine learning

algorithm. In particular, we show that the history-state quantum solution to high-dimensional

linear differential equations takes exactly the form needed to perform quantum analysis of the

solution via quantum machine learning and data analysis. We show how to produce the singular

values and singular vectors of the solution via quantum principal component analysis, and how

to extract the power spectrum of the solution by performing quantum Fourier transforms. The

singular value decomposition and the Fourier analysis reveal the dominant components of the

time evolution, corresponding to large singular values, and to eigenvalues of the transition ma-

trix with small negative real part. Finally we show how to perform quantum wavelet analysis

to reveal rapid transitions and emergent features in the solution at different time scales. These

quantum algorithms for post-processing the solution of linear differential equations can yield an

exponential speedup over classical methods, and could potentially be performed on near term

intermediate scale quantum computers.

The quantum post-processing of the history state to reveal salient features of the history can

be applied to any linear differential equation. To show how quantum post-processing reveals

such features, we focus here on the case of continuous time Markov chains on high-dimensional

spaces, with an emphasis on the spread of disease and opinion in complex social networks. We

have chosen to investigate Markov processes because Monte Carlo methods represent a power-

ful classical method for revealing the features of such processes. Previous quantum algorithms

for Monte Carlo yielded a square root speedup over classical algorithms [5]. By contrast, the

quantum algorithms presented here for analyzing linear dynamics in terms of singular values,

power spectra, and wavelets represent an exponential speedup over existing classical Monte

Carlo methods.

3

Results

Quantum algorithms for linear differential equations Quantum numerical algorithms solve

differential equations by quantizing classical numerical procedures and performing matrix op-

erations on finite, high-dimensional state spaces. We write a general linear differential equation

for a vector in RN with N � 1 in the form

d~x(t)

dt=M~x(t) + ~c (1)

where x(t) represents the state of the system at time t,M is the N × N differential equation

matrix, and c is a forcing term. For example, in the case of Markov models, the state space

is represented by a probability vector x(t) with entries xi(t) indicating the probability of the

system existing in state i at time t, andM is the transition matrix.

Well-known quantum algorithms [6,7] can solve linear differential equations of the form of

Eq. 1, whereM is a sparse matrix, by applying the quantum algorithm for linear systems of

equations [8,9]. The algorithms of Refs. [6,7] are based on a simple underlying idea, but require

highly nontrivial technical improvements in order to achieve a low computational complexity.

For the sake of a clearer exposition, we present here only the basic idea, and refer the reader to

Refs. [6, 7] for a complete presentation of the algorithms. We consider the differential equation

(Eq. 1) for 0 ≤ t ≤ tmax. The idea is based on the standard classical methods for discretizing

Eq. 1 in T time steps each of size h = tmax/T and re-framing it as the solution to an equation

of the form

A~x = ~b (2)

4

where

~x =

~x0~x1~x2...~xT

=~x(0)~x(h)~x(2h)

...~x(Th)

(3)is a ‘vector of vectors’ that contains the values of the state vectors for the system, ~x` = ~x(t`), at

different moments of discretized time t` = `h. A is a matrix that represents the updating action

of the discretized differential operator. The form of A and ~b depends on which discretization

method one employs for the differential equation (e.g., Euler forward, Euler backward, Crank-

Nicolson, etc.). The simplest method is Euler forward, where

A =

I 0 . . . 0 0 0

−(I +M∆t) I . . . 0 0 0. . .

0 0 . . . −(I +M∆t) I 00 0 . . . 0 −(I +M∆t) I

(4)

and

~b =

~x0~c...~c

(5)The solution to the differential equation is then obtained by inverting the matrix in Eq. 2:

~x = A−1~b (6)

Roughly speaking, the quantum algorithms of [6,7] map the classical states onto quantum states,

~x → |x〉, ~b → |b〉, and solve the problem using quantum matrix inversion to construct the nor-

malized version of the unnormalized quantum state |x〉 = A−1|b〉, which represents a quantum

superposition of the solutions of Eq. 1 at different points in time.

We will base our results on the algorithm of Ref. [6], which has the lowest computational

complexity. This algorithm sets T = dtmax ‖M‖e, employs the Taylor expansion of the matrix

5

exponential to solve the differential Eq. 1, and produces a coherent superposition of the quantum

state |x〉 with some garbage state associated to the terms of the Taylor series. We show in the

Methods section that the normalized version |x̄〉 = |x〉/‖|x〉‖ of |x〉 can be recovered from the

quantum state produced by the algorithm of Ref. [6] with O(1) success probability.

Form of the quantum solution The (unnormalized) quantum state |x〉 contains registers for

the states and time-steps:

|x〉 =T∑i=0

|xi〉 |i〉 (7)

where the entries of |xi〉 are the values of the state for timestep |i〉. |x〉 is a quantum ‘history

state’: a superposition of the different timesteps |i〉, correlated/entangled with the corresponding

state vectors |xi〉 at that timestep.

The quantum state in Eq. 7 can now be post-processed using efficient quantum algorithms

such as those for wavelet transforms or quantum machine learning. Such algorithms can pro-

vide useful information about the solution which cannot be obtained efficiently using classical

computation, and we overview these algorithms in later sections.

Alternatively, history states encoding the evolution of an arbitrary quantum circuit at dif-

ferent times can be created by preparing the ground state of a local Hamiltonian [10, 11]. The

quantum post-processing techniques that we apply here to history states of linear differential

equations can also be applied to the history state of the ground state of a local Hamiltonian.

Scaling of the error The computational complexity of the quantum algorithm of Ref. [6] is

linear in the condition number and the time, and logarithmic in the error. We show in the Meth-

ods section that, by running this algorithm and by suitably projecting the generated quantum

state, we can obtain a quantum state that is �-close to the state |x̄〉 in 2-norm with

O

(κ tmax ‖M‖ s poly log

((1 +

tmax ‖~c‖‖~x(0)‖

)κ tmax ‖M‖ sN

�

))(8)

6

elementary quantum gates. Here κ is the condition number of the matrix that is used to diago-

nalizeM and s is the sparsity ofM.

Application to the Schrödinger equation In the case whereM is anti-Hermitian, then Eq.

1 is the Schrödinger equation [10]. In this case, the Fourier analysis of the history state reveals

the eigenvalues and eigenvectors ofM. For example, whenM is the Feynman Hamiltonian,

the history state encodes the history of quantum computation and the Fourier analysis reveals

its eigenstates. Finding the eigenstates of the Feynman Hamiltonian is at least as hard as per-

forming the quantum computation [12].

Continuous time Markov chains To illustrate the power of our methodology, in this paper,

we focus on differential equations for continuous time Markov chains. Markov chains are a

primary tool in mathematics and statistics for modeling the transitions over time of a system

which can exist in one of a set of states. Markov models update probability distributions over

state space by assuming that the probability of making a transition to the next state only depends

on the current state of the Markov chain. Markov chains have applications in many fields

including bioinformatics, population dynamics, and statistical controls [13].

Here, we focus on Markov chains that represent dynamics of complex networks, particularly

those modeling the spread of opinion and disease in social networks. The dynamics of complex

networks are modeled via Markovian techniques by assuming that each node in a network exists

in one of q states. A central challenge in this setup is in handling the dimension of the Markov

state which grows exponentially with the number of nodes in a graph, often rendering the prob-

lem intractable for classical computers. For example, in epidemiology, modeling the dynamics

of infection and recovery for a system of individuals whose interactions make up a complex

network of n nodes is a hard computational problem that involves predicting the behavior of a

very large qn dimensional continuous time probabilistic dynamics [14, 15]. The exact solution

7

for the dynamics of such epidemiological models lies beyond the realm of capability of even the

most powerful classical computers. Consequently, classical approaches to modeling the spread

of epidemics via complex networks of interactions typically rely on various approximations,

such as mean field theory [14].

Continuous time Markov chains are defined by differential equations of the form

d~x(t)

dt=M~x(t) (9)

where ~x(t) is the vector of probabilities for the underlying state of the system at time t, andM

is a matrix of transition rates [16, 17].

Eq. 9 is precisely of the form required in Eq. 1 for efficiently solving differential equations

using quantum algorithms. As shown in the Methods section, the quantum algorithm of Ref. [6]

can be employed to produce a normalized version of the quantum state |x〉 =∑T

i=0 |xi〉 |i〉

storing the state probabilities at discretized times. Note that the quantum algorithm represents

the vector of probabilities as quantum vector of probability amplitudes. Since |x〉 contains the

complete information of the Markov state at different points in time, post-processing algorithms,

which we now outline, can be used to extract useful information from the state.

Generalization to non-Markovian models We can generalize our algorithms to incorporate

prior histories in a “non-Markovian” model. The Markovian nature to the methods for solving

differential equations is reflected in the form of the matrix A in Eq. 4 above. The fact that the

Euler forward method for discretizing a differential equation depends only on the current and

previous state of the system implies that A only has entries on the diagonal and directly below

the diagonal. If we wish to include influences on the present from further in the past (including

the distant past), then we can simply add additional entries to each row: adding a matrix entry

Aij , j < i to the i’th row of A allows the state of the system at time j < i to influence the

updating at time i. This change cannot be directly incorporated in the algorithm of Ref. [6],

8

which relies on the Taylor expansion of the matrix exponential, but it can easily be incorporated

in the previous algorithm of Ref. [7], which directly solves Eq. 2.

Quantum post-processing The solution of our quantum differential equation solver |x〉 exists

in a very high dimensional Hilbert space, and here, we discuss methods to obtain useful infor-

mation from |x〉 using various quantum algorithms that can offer exponential speedups over

classical counterparts. In this study, we focus on the particular case of post-processing out-

puts from continuous time Markov chain models. The algorithms we list here are by no means

comprehensive of the full catalog of algorithms available to quantum computer scientists for

extracting information from these states.

Post-processing: expectation values of quantities The most basic information that can be

extracted from a Markov chain is the expectation value at a given time of a real-valued ob-

servable on the state space. Classically, such expectation values can be estimated efficiently by

Monte Carlo sampling of the Markov chain up to the required time. In the quantum case, ex-

pectation values of quantities can be obtained by estimating the overlap of the history state with

a state encoding the values of the quantity we wish to calculate. In the Supplementary Material,

we consider two quantum algorithms to compute such expectation values. The first is based on

a post-processing of the quantum history state of the Markov chain, and the second is based on

the coherent version of the classical Monte Carlo simulation of the Markov chain. As shown in

the Supplementary Material, one can obtain a quadratic speedup in the error of estimating an

observable using quantum techniques for Monte Carlo sampling.

Post-processing: principal component analysis of data matrix In contrast to the expecta-

tion values of observables, features of the solution such as the singular value decomposition,

the power spectrum, and wavelet analysis cannot be reconstructed efficiently by sampling from

9

the Monte Carlo solution. Here, the power of quantum computation for performing linear alge-

bra on high dimensional spaces provides an exponential speedup over the best known classical

methods.

Principal component analysis of the history state of the differential equation yields useful

information about the solution. First of all, if the history state is effectively low rank, i.e., there

are only a few large singular values, then the description of the time evolution of the Markov

process can be compressed by expressing it in terms of the corresponding singular vectors,

which the quantum principal component analysis also reveals. The history state will be effec-

tively low rank, for example, when the Markov transition matrix has only a few eigenvalues

with small negative parts, so that the dynamics is dominated over longer times by the corre-

sponding eigenvectors. The space spanned by these eigenvectors in turn has high overlap with

the space spanned by the dominant singular vectors.

In the case of continuous time Markov chains, the quantum state in Eq. 7 can be interpreted

as a qn × T data matrix X where each column j corresponds to |xj〉, the probabilities of the

Markov state at timestep j. The dominant singular values and corresponding singular vectors

of this matrix can be extracted from |x〉 by performing quantum principal component analysis

(qPCA) on the |xj〉 and |j〉 registers which runs in O(Rn log q) time where R is the rank of

X [2]. Note, that qPCA in this setting is equivalent to performing a Schmidt decomposition on

the Hilbert spaces spanned by registers |xj〉 and |j〉. The qPCA performs this decomposition

via density matrix exponentiation [2]. It is often the case that the effective rank R (the number

of large singular values) of X is small with respect to the number of states qn, and later, we

show that the effective rank is in fact very small for the example models we consider. Note that

because our method acts directly on a quantum state, quantum inspired algorithms for PCA do

not have access to the data structure for extracting the singular vectors and singular values of

the history state [18–20]. Specifically, in the Supplementary Material, we show that classical

10

Monte Carlo methods cannot efficiently extract the singular vectors and the singular values of

the history state (Eq. 7) whenever the support of the probability distribution is exponentially

large in the number of nodes of the network.

After performing qPCA via density matrix exponentiation on copies of |x〉, we have a de-

composition of the data matrix into left and right singular vectors:

qPCA : |x〉 →T∑j=0

|lj〉 |rj〉 |σ̃j〉 , (10)

where |lj〉 are the left singular vectors corresponding to the Markov states, |rj〉 are the right

singular vectors corresponding to the temporal states, and |σ̃j〉 are estimates of the singular val-

ues. The singular values |σ̃j〉 represent the weight of the left (Markov state) and right (temporal

state) singular vectors in the solution. It is conventional to take the ordering in j in Eq. 10 to be

from the largest to smallest singular values.

The left singular vectors |lj〉 can be interpreted as the most common profile of Markov

states. The first left singular vector corresponds to the profile with the greatest contribution to

the data matrix, often the steady state of the Markov process. The next few singular vectors

typically correspond to the profile of states in the early progression of the Markov simulation

before steady state is achieved (see simulations later for examples).

The right singular vectors |rj〉 detail the progression of the corresponding left singular vec-

tors over time. For example, the first singular vector is typically weak during the early progres-

sion and grows to a constant value as the steady state arises. The next few right singular vectors

show when the corresponding left singular vectors take prominence, often highest in magnitude

at early points in time (see simulations later for examples).

Decomposing the data into singular vectors also allows one to apply efficient transforma-

tions to the singular vectors using quantum post-processing methods. For example, if one is

interested in analyzing the Markov states in the frequency domain, a quantum Fourier trans-

11

form can be applied to the right singular vectors. Later, in our example, we show that the

dominant singular vectors correspond to slowly varying dynamics at low frequencies and the

later singular vectors correspond to more rapidly varying dynamics at higher frequencies.

We noted above that when the Markov transition matrix has only a few eigenvalues with

small negative parts, the history state will be effectively low-rank, with only a few large sin-

gular values. The quantum singular value analysis can be used to probe other aspects of the

history as well. For example, we can perform quantum principal component analysis on a slid-

ing window within the history: as this window moves forward, a fundamental change in the

dynamics of the process can have a signature in changes in the singular values, and in the form

of the corresponding singular vectors. These changes can be revealed by performing swap tests

between the dominant singular vectors at different points during the evolution.

Post-processing: efficient quantum transformations Fourier transforms and wavelet trans-

forms are commonly used in the analysis of large datasets, especially time series [21–24]. Dis-

crete wavelet transforms, for example, can be used to identify statistical patterns in a time series.

With a quantum computer, Fourier transforms and certain discrete wavelet transforms can be

performed exponentially faster than their classical counterparts [25–27]. These transforms can

be applied to the data contained in our output quantum state (Eq. 7). For example, a Fourier

transform or wavelet transform can be applied to the time register, e.g. to observe the data in the

frequency domain or to compress the data in terms of the dominant wavelets. Let Ujk = 〈k|j〉

be the element of the unitary matrix U that maps the states |j〉 to the transform states |k〉 (fre-

quency states in the case of the quantum Fourier transform; wavelets in the case of the discrete

wavelet transform). Applying U to the temporal register, we obtain the state

T∑j=0

|xj〉U |j〉 =T∑

j,k=0

|xj〉Ujk |k〉 =T∑k=0

|yk〉 |k〉 , (11)

12

where |yk〉 =∑T

j=0 Ujk |xj〉, is the state of the system correlated with the kth frequency or

wavelet state in the temporal register. Sampling from the temporal register then yields the dom-

inant frequencies/wavelets, and the spatial register yields the state of the system correlated with

those frequencies/wavelets. For example, as noted above in the discussion of the Schrödinger

equation, in Eq. 1, if ~c is 0 and the matrixM is anti-Hermitian, performing a quantum Fourier

transform on the temporal register yields the purely complex eigenvalues ofM and the output

contains the corresponding eigenvectors [28]. More generally, when the eigenvalues ofM have

both real and imaginary components, performing the quantum Fourier transform reveals the

power spectrum of the solution: a complex eigenvalue a + ib manifests itself as a Lorentzian

peaked at the natural frequency√a2 + b2.

By contrast, classical Monte Carlo sampling does not obviously extract the proper informa-

tion for performing Fourier or wavelet transforms on the quantum state (see section on classical

algorithms for principal component analysis in the Supplementary Material).

Post-processing: quantum machine learning In the past few years, many quantum algo-

rithms for machine learning have been proposed that can be performed exponentially faster

than classical counterparts in cases where data is input as a quantum state [1, 3, 4, 29–33]. Us-

ing quantum algorithms for continuous time Markov chain models proposed here, input states

for these algorithms are generated efficiently thus preserving the exponential speedup of these

algorithms even in the construction of input states.

When data in the form given by Eq. 7 is input into machine learning algorithms, applications

of machine learning algorithms are numerous. Here we list some of these applications, grouped

based on the type of model that can be used. First, quantum models have been proposed for

compression of data or efficient readout. These models include quantum auto-encoders [33,34]

and qPCA as discussed before [2]. Second, a wide range of algorithms implementing kernel

13

methods can be used to classify data, identify key features in the data, or measure similarity

between different datasets [4, 35–37]. Indeed, if we trace out the temporal register in Eq. 7,

the state register is described by the (unnormalized) density matrix∑

j |xj〉 〈xj|, which is the

covariance matrix for the synthetically generated data; similarly, if we trace out the state register,

the temporal register is described by the density matrix∑

ij〈xi|xj〉 |j〉 〈i|, which is the kernel

matrix for the data. That is, the quantum differential equation solver gives us direct access to the

states required to attain the potential exponential speedups of quantum kernel methods. Third,

output states can be inputted into parameterized quantum circuits or quantum neural networks

to identify key features or perform machine learning tasks [31, 38, 39].

Example simulation for epidemic processes The analysis of epidemic spreading – viral or

social – is often modeled as a dynamical process on a complex network [14]. Theoretical

approaches to epidemic processes typically assume transitions (e.g. rates of infection) occur

as Poisson processes which correspond to models of continuous time Markov chains [14, 40].

Classically, numerical simulation of continuous time Markov processes is intractable for large

networks as the dimension of the Markov state grows exponentially with the number of nodes

in the network or graph.

To demonstrate the applicability of our quantum algorithm, we simulate continuous time

Markov chain models on simple seven node networks. We choose a relatively small network

so that we can still visually represent the full solution to the continuous time Markov chain.

Here, we present models both for analyzing viral and social epidemics. Also, for the sake

of brevity and ease of graphical presentation, we implement susceptible-infected-susceptible

(SIS) epidemiological models which have only two states per node: susceptible and infected.

Nodes can become infected multiple times in this model – i.e., there is no recovered state as in

a susceptible-infected-recovered (SIR) model. For contagion on social networks, we consider

14

models where nodes can exist in one of three states labeled liberal, conservative, and undecided.

Epidemic simulations of viral contagion We present analysis of a Markov simulation for

a susceptible-infected-susceptible (SIS) model on a single network shown in Fig. 1A. Our

simple model features many common properties of continuous time Markov chain simulations.

Notably, it is common that Markov transition matrices have only a small number of dominant

eigenvalues (i.e., those whose values are close to zero) thus rendering them suitable for analysis

similar to that performed here for small networks. Of course, network models with more nodes

will likely have emergent phenomena that will not appear in this small network – phenomena

that one may hope to analyze using quantum computers [14, 40].

For an SIS model, the full state is described by a vector ~x of length 2n (n = 7 for our

example). All transitions are modeled as Poisson processes with transition matrix Q.

d~x

dt= Q~x (12)

We begin in an initial state where any node has a 35% probability of being infected and con-

duct analysis over the intermediate phase of the epidemic, between days one and two. Specif-

ically, during that period, we numerically construct the history state of the Markov simulation

using a Forward Euler method that begins at the state of the epidemic on day one and ends at

day two over T = 1027 timesteps (step size h ≈ 0.001 days):

~xt+1 = ~xt + hQ~xt (13)

From day one to day two, the epidemic has spread sufficiently that multiple individuals are

likely to be infected, and the probability distribution has spread throughout the space. As noted

above, this is the regime where the quantum algorithm provides an exponential advantage over

existing classical Monte Carlo techniques.

15

0

1

2

3

4

5

6

A

states 0.0 0.2 0.4 0.6 0.8 1.0time (in days)

Bprobabilities of states

10 3

10 2

10 1

100

Figure 1: (A) The 7 node network used for simulation of the continuous time Markovchain. Line widths correspond to the rate of infection from infected neighbor nodes (rSI ={0.4, 0.8, 1.6} from thinnest to thickest lines). Network has the feature that some nodes (e.g.,node 2 and 3) are strongly connected to neighboring nodes and others (e.g., node 5) are not.(B) Probabilities of Markov states over time shown as a colorbar chart (logarithmically scaled).States are enumerated as rows on the left hand side, each denominated by a 7 node colorbarnumbered node 0 on the left to node 6 on the right. Dark/light color indicates a node in thatstate is infected/susceptible respectively.

16

Fully simulating the above for T steps constructs a data matrix X where column i is equal

to xi−1 (we assume initial state x0 is included in this matrix as well).

X =

| | |~x0 ~x1 . . . ~xT| | |

(14)where we note that this data matrix can be interpreted as a classical version of the quantum

output state shown in Eq. 7.

To construct the matrix Q, we use the method detailed in [41]. We assume transitions from

infected to susceptible (i.e., recovery rate) occur at rate rIS = 0.33 indicating that it takes about

three days on average to recover from infection. Transitions from susceptible to infected occur

at a rate rSI ∈ {0.4, 0.8, 1.6} depending on a node’s connection strength to a neighboring node

(see Fig. 1A caption).

In Fig. 1B, we plot the probabilities of states of the Markov process over time providing

a visualization of the data matrix X . For larger networks, visualization of this data matrix

cannot be efficiently performed and we now turn our attention to efficient quantum algorithms

for extracting salient features from this data matrix.

Simulations: quantum principal component analysis (qPCA) The output of our contin-

uous time Markov chain algorithm is stored in a data matrix, visualized in Fig. 1B. In the

quantum setting, this data matrix is a quantum state which we can subsequently post-process

using various efficient quantum algorithms. One available post-processing algorithm is quan-

tum Principal Component Analysis (qPCA) as in Eq. 10, which can compress the data into its

singular vectors [2]. If the matrix is low rank, as in our example with exponentially decaying

singular values (see Fig. 2A), qPCA can be performed in time logarithmic in the dimension of

the full Markov state [2]. The principal components can be subsequently transformed or even

measured.

17

0 20 40 60 80 100 120singular value index

10 14

10 11

10 8

10 5

10 2

101

singu

lar v

alue

A

states 1.0 0.5 0.0 0.5singular vector value (scaled)

B

Vector 1Vector 2Vector 3Vector 4

0.0 0.2 0.4 0.6 0.8 1.0time (in days)

0.08

0.06

0.04

0.02

0.00

0.02

0.04

singu

lar v

ecto

r val

ue

C

Vector 1Vector 2Vector 3Vector 4

Figure 2: (A) Singular values of data matrix X decay exponentially fast. (B) First four leftsingular vectors scaled by the square root of their corresponding singular values show that muchof the disease progression can be understood by just observing the first few vectors. States areenumerated as rows on the left hand side, each denominated by a 7 node colorbar numberednode 0 on the left to node 6 on the right. Dark/light colors indicate a node in that state isinfected/susceptible respectively. (C) Values of the right singular vectors scaled by the squareroot of their corresponding singular values show the progression of the epidemic over time.The first singular vector depicts the steady state and the next few singular vectors detail theintermediate course of the Markov process.

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The most dominant left (Fig. 2B) and right (Fig. 2C) singular vectors can be interpreted

as the most common profile of states (left vectors) and their corresponding trajectories in time

(right vectors). In our example, the first singular vector corresponds to the steady state of our

epidemic. Note that it takes prominence almost completely throughout the course of the simula-

tion (see right singular vector in Fig. 2C). The second singular vector plots important changes in

the epidemic as more nodes become infected. The corresponding right singular vector plots the

steady, almost linear, transition over time as this singular vector takes prominence. Similarly,

the third and fourth singular vectors plot trends in the progression of the epidemic, especially

in early phases where nodes transition to recovery or infection.

Simulations: Fourier and wavelet analysis For small networks such as the one studied here,

the data in the Fourier domain is dominated by the steady state contributions as shown in Fig.

3A. With quantum algorithms, one also has the option of transforming the individual singular

vectors into their frequency components as shown in Fig. 3B. The second to fourth singular

vectors all have strong contributions from low frequencies, whose values provide an indication

of the rate of change in the progression of the Markov chain. Given the small network size, this

dominance of low frequency components is perhaps not altogether surprising. Larger networks

encounter phenomena not observed in small networks, and may potentially reveal interesting

features in the Fourier domain [14, 40] if they are analyzed with a quantum computer.

Quantum computers offer the advantage of efficient post-processing via wavelet transforms.

Here, we transform the time dimension of our data matrix using a Haar wavelet transform, which

can be performed efficiently on a quantum computer [25]. When viewed in the Haar wavelet

basis (form of wavelets shown in Fig. 4A), one can analyze the characteristic timescales over

which differences in the Markov state probabilities become apparent. Perhaps unsurprisingly,

as shown in Fig. 4B, the zeroth Haar vector is most prominent as this corresponds to the steady

19

10 5 0 5 10Frequency (1/days)

10 6

10 4

10 2

100pr

obab

ility

of m

easu

rem

ent A

10 5 0 5 10Frequency (1/days)

10 4

10 3

10 2

10 1

100

singu

lar v

ecto

r val

ue

B

Vector 1Vector 2Vector 3Vector 4

Figure 3: (A) Measurements made in the Fourier domain (plotted only for low frequency com-ponents) have high probability in the zero frequency (steady state) and low frequency compo-nents. (B) The first singular vector, plotted in the Fourier domain, is dominated by the zerofrequency component corresponding to the steady state. Later singular vectors take prominencein low frequency components which indicate the rate of change during the intermediate pro-gression of the Markov chain.

20

state of the Markov chain. More interesting results are observed in analysis of the singular

vectors, which can also be transformed into the Haar domain as shown in Fig. 4C. Here, clear

differences can be observed in the Haar basis of the steady state singular vector (first singular

vector) and later vectors. The first singular vector is dominated by the zeroth Haar wavelet

(constant wavelet) since that singular vector corresponds to the steady state. The next few

singular vectors corresponding to changes in the intermediate progression of the epidemic are

dominated by Haar vectors with support over various phases. For example, the third and fourth

singular vectors take large values over Haar vectors with support in the early phases of the

simulation (e.g., fourth and eighth Haar vectors).

Epidemic simulations of social opinion Continuous time Markov chains can also be imple-

mented to simulate the spread of social opinion. Here, we consider a model where nodes can

exist in one of three states: conservative, liberal, or undecided. As in our analysis on viral epi-

demics, we perform simulations on the same 7-node network (see figure 1A). Similar to before,

transitions from undecided to liberal or conservative occur at rates dependent on the strength

of connection to other liberal or conservative nodes respectively. The data matrix is analyzed

between days one and two of the ”social epidemic”, where at day zero, all states are equally

likely.

As shown in figure 5A, the data matrix in this case is similarly low rank. Furthermore, we

see similar progressions over time in the right singular vectors as shown in figure 5B. The first

singular vector corresponds to steady state contributions, whereas later singular vectors chart

the most prominent changes in the data matrix over time.

21

0 21 22 23 24 25 26Haar wavelet number

10 9

10 7

10 5

10 3

10 1

prob

. of m

easu

rem

ent (

log

axis) B

0 21 22 23 24 25 26Haar wavelet number

1.00

0.75

0.50

0.25

0.00

0.25

0.50

singu

lar v

ecto

r val

ue

C

Vector 1Vector 2Vector 3Vector 4

0th

4th

8th

12th

1st

5th

9th

13th

2nd

6th

10th

14th

3rd A

7th

11th

15th

Figure 4: (A) Each Haar wavelet has support over a characteristic timescale indicated by thewavelet number. Haar wavelet numbers between powers of 2 take the same form and are offsetfrom each other in the time dimension. As a visual aid, we show here the first 16 discrete Haarwavelets. (B) Quantum state has high probability in the zeroth Haar wavelet (steady state).(C) Wavelet transform of singular vectors shows that the first singular vector is strongest in thezeroth Haar vector (steady state). Later singular vectors take prominence in Haar vectors withsupport in the early stages of epidemic.

22

0 200 400 600 800 1000singular value index

10 15

10 12

10 9

10 6

10 3

100

singu

lar v

alue

A

0.0 0.2 0.4 0.6 0.8 1.0time (in days)

0.04

0.03

0.02

0.01

0.00

0.01

0.02

singu

lar v

ecto

r val

ue

B

Vector 1Vector 2Vector 3Vector 4

Figure 5: Quantum principal component analysis of the spread of social opinion. (A) Singularvalues of data matrix X decay exponentially fast. (B) Values of the right singular vectors scaledby the square root of their corresponding singular values show the progression of social opinionover time. The results are consistent with prior results for the analysis of viral epidemics on thesame network.

23

Discussion

Common quantum algorithms for solving linear differential equations output quantum states

corresponding to solutions of physical models in high dimensional vector spaces. These out-

put states store the complete history of the solution to a differential equation, allowing one to

perform efficient quantum post-processing on these solution states. Our approach avoids a com-

monly cited drawback of many quantum machine learning and signals processing algorithms –

that classical inputs must be mapped into quantum states or stored in qRAM. The quantum

linear differential equation solver efficiently produces as outputs states of exactly the form re-

quired for quantum machine learning and signals processing algorithms. We focus here on the

case of Markov models and propose efficient quantum algorithms for evaluating continuous

time Markovian and non-Markovian models. Our algorithms allow for efficient simulation of

Markov models on the complete state of a Markov chain. Outputs of our models are quantum

states which can be efficiently generated and then passed as inputs for other efficient quantum

post-processing algorithms (e.g., quantum signal analysis and machine learning). The quantum

post-processing reveals features of the data such as the singular value decomposition, the power

spectrum, and wavelet decompositions, which cannot be reconstructed efficiently using classi-

cal sampling algorithms. When applied to complex networks, the quantum algorithms may be

used to reveal such fundamental features of the dynamics of epidemics potentially exponentially

faster than classical algorithms.

Materials and Methods

Generation of the quantum history state

Let T = dtmax ‖M‖e, let δ > 0 and let k ∈ N with k ≥ 5 and (k + 1)! ≥ 2T . The quantum

algorithm of Ref. [6] discretizes the differential equation (Eq. 1) in T time steps of size h =

24

tmax/T , truncates the Taylor expansion of the matrix exponential at the k-th order and produces

an approximate normalized version |φ〉 of the quantum state

|y〉 =T−1∑i=0

k∑j=0

|i, j〉|yi,j〉+ |T, 0〉|yT,0〉 . (15)

For sufficiently large k, the vectors |yi,0〉 are close to the solution of the linear differential

equation (Eq. 1) at the i-th time step [6, Theorem 6]:

‖~x(ih)− |yi,0〉‖ ≤ 2.8κ i‖~x(0)‖+ tmax ‖~c‖

(k + 1)!, (16)

and the vectors |yi,j〉 for j ≥ 1 are some garbage vectors associated to the terms of the Taylor

expansion of the matrix exponential, of which we do not need the exact form. Their norms are

upper bounded by [6, eqs. (99), (100)]

‖|yi,j〉‖ ≤‖|yi+1,0〉‖+ ‖|yi,0〉‖

(3− e) j!. (17)

Let

|ȳ〉 = |y〉‖|y〉‖

(18)

be the normalized version of |y〉. The quantum state |φ〉 satisfies

‖|φ〉 − |ȳ〉‖ ≤ δ (19)

and the algorithm requires

O

(κ k2 tmax ‖M‖ s poly log

κ k tmax ‖M‖ sNδ

)(20)

elementary quantum gates [6, eq. (128)].

An approximation of the quantum history state |x〉 of Eq. 7 can be obtained by projecting

the second register of the quantum state |φ〉 on the 0 value. Let 〈0|φ〉 be the unnormalized

projection. In the following, we show that its success probability is O(1), and that choosing

δ = O(�) , k = O

(log

((1 +

tmax ‖~c‖‖~x(0)‖

)κ tmax ‖M‖

�

))(21)

25

we can achieve ∥∥∥∥ 〈0|φ〉‖〈0|φ〉‖ − |x̄〉∥∥∥∥ ≤ � . (22)

From Eq. 20, this modification of the algorithm of Ref. [6] requires

O

(κ tmax ‖M‖ s poly log

((1 +

tmax ‖~c‖‖~x(0)‖

)κ tmax ‖M‖ sN

�

))(23)

elementary quantum gates.

Success probability

We assume that

δ ≤ 12√

66. (24)

We have from Eq. 17

k∑j=1

‖|yi,j〉‖2 ≤(‖|yi+1,0〉‖+ ‖|yi,0〉‖)2

(3− e)2∞∑j=1

1

j!2≤ 1.28 ∗ 2 ‖|yi+1,0〉‖

2 + ‖|yi,0〉‖2

(3− e)2, (25)

andT−1∑i=0

k∑j=1

‖|yi,j〉‖2 ≤1.28 ∗ 4(3− e)2

T∑i=0

‖|yi,0〉‖2 ≤ 65T∑i=0

‖|yi,0〉‖2 . (26)

Let 〈0|y〉 be the projection of |y〉 onto the 0 value of the second register. The success probability

of such projection is

‖〈0|ȳ〉‖2 =∑T

i=0 ‖|yi,0〉‖2∑T

i=0 ‖|yi,0〉‖2 +

∑T−1i=0

∑kj=1 ‖|yi,j〉‖

2≥ 1

66. (27)

Let 〈0|φ〉 be the projection of |φ〉 onto the 0 value of the second register. We have from Eq. 19,

27 and 24

‖〈0|φ〉‖ ≥ ‖〈0|ȳ〉‖ − ‖〈0|ȳ〉 − 〈0|φ〉‖ ≥ ‖〈0|ȳ〉‖ − δ ≥ 12√

66, (28)

therefore the success probability of the projection satisfies

p = ‖〈0|φ〉‖2 ≥ 1264

. (29)

26

Error analysis

From Eq. 16, the distance between the quantum history state |x〉 and the projection 〈0|y〉

satisfies

‖|x〉 − 〈0|y〉‖2 =T∑i=0

‖~x(ih)− |yi,0〉‖2 ≤ 2.82 κ2(‖~x(0)‖+ tmax ‖~c‖)2

(k + 1)!2T (T + 1) (2T + 1)

6,

(30)

then

‖|x〉 − 〈0|y〉‖ ≤ κ‖~x(0)‖+ tmax ‖~c‖2k+4

√3T (T + 1) . (31)

We have from Lemma 1∥∥∥∥|x̄〉 − 〈0|y〉‖〈0|y〉‖∥∥∥∥ ≤ ‖~x(0)‖+ tmax ‖~c‖‖|x〉‖ κ

√3T (T + 1)

2k+3≤(

1 +tmax ‖~c‖‖~x(0)‖

)κ√

3T (T + 1)

2k+3.

(32)

We choose

k =

⌈log2

((1 +

tmax ‖~c‖‖~x(0)‖

)κ√

3T (T + 1)

4�

)⌉

= O

(log

((1 +

tmax ‖~c‖‖~x(0)‖

)κ tmax ‖M‖

�

)), (33)

such that ∥∥∥∥|x̄〉 − 〈0|y〉‖〈0|y〉‖∥∥∥∥ ≤ �2 . (34)

We have from Eq. 19, 27 and Lemma 1 again∥∥∥∥ 〈0|φ〉‖〈0|φ〉‖ − 〈0|ȳ〉‖〈0|ȳ〉‖∥∥∥∥ ≤ 2 ‖〈0|φ〉 − 〈0|ȳ〉‖‖〈0|ȳ〉‖ ≤ 2√66 δ , (35)

such that choosing

δ =�

4√

66(36)

we have ∥∥∥∥ 〈0|φ〉‖〈0|φ〉‖ − 〈0|ȳ〉‖〈0|ȳ〉‖∥∥∥∥ ≤ �2 (37)

27

and ∥∥∥∥ 〈0|φ〉‖〈0|φ〉‖ − |x̄〉∥∥∥∥ ≤ ∥∥∥∥ 〈0|φ〉‖〈0|φ〉‖ − 〈0|ȳ〉‖〈0|ȳ〉‖

∥∥∥∥+ ∥∥∥∥ 〈0|y〉‖〈0|y〉‖ − |x̄〉∥∥∥∥ ≤ � (38)

as required.

Lemma 1. Let v, w be vectors in a normed vector space. Then,∥∥∥∥ v‖v‖ − w‖w‖∥∥∥∥ ≤ 2 ‖v − w‖‖v‖ . (39)

Proof. We have ∥∥∥∥ v‖v‖ − w‖w‖∥∥∥∥ ≤ ‖v − w‖+ |‖w‖ − ‖v‖|‖v‖ ≤ 2 ‖v − w‖‖v‖ . (40)

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32

Supplementary Material

Sampling and calculating observables of Markov states

To determine the Tqn individual probabilities of a Markov state xj(k), for j = 0 to T , k = 1 to

N = qn stored in a quantum state, one would have to make an exponentially large number of

measurements. However, one can use quantum measurements to reveal a wide variety of desired

properties of the quantum system at time j. To extract the expectation value of some desired

quantity Q (total number infected, variance of the infection rate across the graph, existence of

hot spots, etc.) we need to make a quantum measurement to estimate the expectation value

〈Q〉 =N∑k=1

xj(k)Q(k) (41)

where Q(k) is the value of Q on the k’th state of the vertices of the graph.

First, we have to make sure that we obtain the state |~xj〉 with high probability. In the

formulation given above, this state only occurs in the overall superposition |x〉 with amplitude

O(1/√T ). The standard way to amplify the probability of finding the answer at the desired

time j [7] is to pad out the matrix A following the j’th row with O(T ) rows of the form

(0 . . . 0 − I I 0 . . . 0) (42)

where the I term in each −I I 0 sequence lies on the diagonal. These rows induce a trivial

dynamics in which all the states in the solution following the j’th state also contain the state

|~xj〉. This technique allows us to obtain the state |~xj〉 with probability O(1).

To obtain 〈Q〉 =∑N

k=1 xj(k)Q(k), we use standard techniques of quantum state preparation

[42]. We assume that we are given access in quantum superposition to the individual values of

the variable Q(k) together with its partial sums over ranges of k; The techniques of [42] then

33

allow us to construct the (unnormalized) state

|Q〉 =N∑k=1

Q(k)|k〉 (43)

in time O(logN) = O(n log q). Define ZQ =∑N

k=1Q(k)2. Even if ZQ is not known before-

hand, its value is revealed during the state preparation process [42]. The normalized version of

|Q〉 is then

|Q̃〉 = Z−1/2Q |Q〉 (44)

We can now use a swap test between |Q̃〉 and |x̃j〉 to measure the overlap 〈Q̃|x̃j〉. This

overlap in turn allows us to measure

〈Q〉 =N∑k=1

xj(k)Q(k) = 〈Q|xj〉 = Z1/2j Z1/2Q 〈Q̃|x̃j〉 (45)

In conclusion, even though we don’t have access to the individual probabilities for states, the

quantum algorithm allows us to measure expectation values for a wide variety of observables

efficiently. This method allows us to use the quantum algorithm to extract expectation values of

the desired quantities.

Comparison to classical complexity Extracting expectation values could also be done clas-

sically using classical Monte Carlo to sample from the probabilistic dynamics. Because of the

local form of the probabilistic updating rule, the number of computational steps required to

draw one sample of the Markov chain at time t scales as

O(n(t/h) log q) . (46)

The average of Q over O(

log 1δ

/�2)

samples is �-close to 〈Q〉 with probability at least 1 − δ,

and its computation has complexity

O

(n(t/h) log 1

δlog q

�2

). (47)

34

Quantum speedup of Monte Carlo sampling The quantum algorithm of Ref. [5] provides a

quadratic improvement in the dependence of the complexity Eq. 47 of Monte Carlo sampling on

the precision �. More precisely, let us assume that 0 ≤ Q(k) ≤ 1 for any k = 1, . . . , N (this can

always be achieved by a suitable linear redefinition of Q). Let U be a quantum unitary operator

that implements a unitary dilation of the classical algorithm for the Monte Carlo sampling of

the Markov chain. We can assume that the complexity of U has at worst a constant overhead

with respect to the classical algorithm, and therefore has the same scaling Eq. b46. Then,

Theorem 2.3 of [5] implies that, for any 0 < � < 1 and any 0 < δ < 1 there exists a quantum

algorithm that, with O(

log 1δ

/�)

applications of U , outputs µ ∈ R such that |µ− 〈Q〉| < �

with probability at least 1− δ. The complexity of the algorithm is therefore

O

(n(t/h) log 1

δlog q

�

), (48)

with the promised quadratic improvement in the �-dependence with respect to Eq. 47.

Classical algorithms for Principal Component Analysis

Here we show that the singular vectors and singular values of the data matrix X cannot be effi-

ciently estimated with classical methods whenever ‖|xj〉‖2 is exponentially small in the number

of nodes for any time step, i.e., if the size of the support of the probability distribution is al-

ways exponential. More precisely, we show that none of the entries of X†X can be estimated

efficiently. Indeed, if xj(k) is the probability of the k-th state at the time step j, the jj′ entry of

X†X is (X†X

)jj′

=∑k

xj(k)xj′(k) , (49)

and is equal to the probability that, in a couple of independent Monte Carlo simulations of

the Markov chain, the state of the first simulation at the step j is equal to the state of the

second simulation at the step j′. This probability can be estimated by running many couples

35

of simulations of the Markov chain. However, the estimate will be zero until a couple with the

state of the first simulation at the step j equal to the state of the second simulation at the step j′

is found. This event will typically happen after

O

(1

(X†X)jj′

)≥ O

(1

maxi ‖|xi〉‖2

)(50)

runs, which is exponentially large in the number of nodes if ‖|xj〉‖2 is exponentially small for

any time step.

Additional figures and simulations

Supplementary to the main text, we show results here for a simulation of a Markov chain which

incorporates effects of ”social distancing” in the Markov model. Specifically, we simulate a

viral epidemic on the same network as in the main text where transitions from susceptible to

infected and vice-versa occur with rates rSI = 1.5 and rIS = 0.33 respectively as long as

three or fewer nodes are infected. When four or more nodes are infected, ”social distancing” is

enacted and transitions from susceptible to infected occur at a fifth of the original rate (rSI =

0.3). As expected, this shifts the steady state away from situations where all nodes are infected

to those where four nodes are infected.

The complete progression of this model is plotted in Fig. 6B. Note, that the state where

all nodes are infected is now unlikely; instead, the states where four or five nodes are infected

become very likely (i.e., the social distancing works).

As with the original model, singular values decay exponentially rapidly (see Fig. 7A). The

first four left singular vectors are plotted in Fig. 7B scaled by their corresponding singular

value. The steady state singular vector has clearly changed with respect to the original model.

Analysis of the first singular vector shows that the dominant states are those where no node is

infected and four nodes are infected (i.e., the transition point of social distancing).

36

0

1

2

3

4

5

6

A

states 0.0 0.2 0.4 0.6 0.8 1.0time (in days)

Bprobabilities of states

10 3

10 2

10 1

100

Figure 6: (A) The 7 node network used for simulation of continuous time Markov chain wheretransitions from susceptible to infected and vice-versa occur with rates rSI = 1.5, reduced bya factor of five when four or more nodes are infected. (B) Probabilities of Markov states overtime shown as colorbar chart (logarithmically scaled). States are enumerated as rows on the lefthand side, each denominated by a 7 node colorbar numbered node 0 on the left to node 6 on theright. Dark/light color indicates a node in that state is infected/susceptible respectively.

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0 20 40 60 80 100 120singular value index

10 14

10 11

10 8

10 5

10 2

101

singu

lar v

alue

A

states 1.00 0.75 0.50 0.25 0.00singular vector value (scaled)

B Vector 1Vector 2Vector 3Vector 4

0.0 0.2 0.4 0.6 0.8 1.0time (in days)

0.06

0.04

0.02

0.00

0.02

singu

lar v

ecto

r val

ue

C

Vector 1Vector 2Vector 3Vector 4

Figure 7: (A) Singular values of data matrix X decay exponentially fast. (B) First four leftsingular vectors scaled by the square root of their corresponding singular value show that stateswhere four nodes are infected become prominent as this is the inflection point for social dis-tancing. States are enumerated as rows on the left hand side, each denominated by a 7 nodecolorbar numbered node 0 on the left to node 6 on the right. Dark/light colors indicate a node inthat state is infected/susceptible respectively. (C) Values of the right singular vectors scaled bythe square root of their corresponding singular value show the progression of the epidemic overtime. The first singular vector depicts the steady state and the next few singular vectors detailthe intermediate course of the Markov process.

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base model with people nodes

state: ۧȁ𝑃

model with variable connection nodes

In SIS model, each node either susceptible or infected

state: ۧȁ𝑃 ۧȁ𝐶

Connection nodes mediate Markov process and are part of state (e.g., can be on/off as two states)

model with testing/treatment nodes

state: ۧȁ𝑃 ۧȁ𝐻

treatment nodes can e.g., have on/off state where on state has faster recovery for connected nodes

model with different population segments

state: ൿห𝑃1 ൿห𝑃2

e.g., can have a separate segment for older vs. younger population, each with different recovery time

Figure 8: By customizing the properties of nodes in a network, Markov states can model variousdifferent phenomenon. Here, we show some of the options available to expand the function-ality of a Markov state. In these cases, states are stored in a tensor product structure whereinformation corresponding nodes of different types can be stored in separate registers.

Markov states incorporating more than just simple nodes

Markov models can incorporate nodes of different types which interact in a customized fash-

ion. Fig. 8 outlines some of the different options available to one modeling epidemiological

processes. Utilizing quantum algorithms, nodes of different types can be stored in separate

registers. Analysis and post-processing of the Markov states using a quantum state can take

advantage of the structure inherent in these expanded models.

Computational details

All simulations were performed in Python. Code is available upon request.

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